Elements of Electromagnetic Theory [1 ed.]

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ELF.11\ENTS OF ELECTROMAGNETIC THEORY

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ELEMENTS OF

ELECTROMAGNETIC THEORY

S.

J. BARNETT, PH . D.

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3'rbr !lork THE MACMILLAN COMPANY LON"DON : MACMILLAN & CO., LTD.

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BT T K E NAt: NILLAN COMPANY

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DEDICATED

GRATITUDE ANU AFFF..CTIO?f

PROFESSOR FRANCIS H. SMITH, LL.D.

U~IVERSITV

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VIRGINIA

PREFACE. In this treatise I have tried to present in systematic and definite form a simple, rigorous, and thoroughly modern introduction to the fundamental principles of electromagnetic theory, together with some of the simpler of th~r more interesting and important non-tcchnica] applications. The work makes no pretense to completeness, but is written for the serious student of physic..-.. who will make liberal use of more detailed treatises, of hand-books, and of jouma1s, as occasion demands. I am of course indebted to many books and memoirs. My obligations arc especially great, as the most cursory examination of the book will show, to the works of Maxwell, Heaviside, and Poynting. I am also much indebted to Professor A. G. Webster for the use of a number of excellent diagrams from his treatise on electrical theory. 5. j. IlAKNETT, STA ~J'O llll U!111YhSrTY 1 C,1.u,01.,ru,

Jun~,

1903,

CONTENTS. c....rru. I. 11.

PAGL

GDEaAL :CLECTllOSTATIC THEOR Y •••••••• , •••••••••• ,... . . .. . . .. . . .

lUEAL ELECTRIC

FIELDS ASI>

CosuE.....Sll!:llS WITH

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llo•o-

GF-'l:WUS DIELECTRJCS.. .... . ............ . ... . ........ ~•·····•··•····•

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STANDA kO CotrnENSEJlS.

JV.

El.Ecr1uc F JELDS W1TH Two Oil MOIIE DIELECTRICS ..• ••. .•• 1 39

COXDESSEll $\'STEMS.

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ET EJlS. •• ••• ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• 122 V.

\1. VII.

REVE&SlllLE TH ER MA L

ELECTRIC ABsokJ'TION.

EJPl"ECT.

El,ECTROSTRICTIOS •••• ••••• 168

Et.EC'UETS . ••• , ••• . , • ..•. , • .. .•• . .• . •.• •. 176

CO K PU ISON OP O I ELEC'Ta.JC CONSTA!ffS.

SPF.C l l'IC

bmvc-

TIVE CAPACITY •••• •• •• •• • • . •••••• ••• • ••• •• •• •• •••••••• ••• • •• •• ,, • . •• , •• 192

vm.

TH £

ELECTAIC CONDUCTIOS CUJlkF.1'ol.

ISTklSSIC

Eu~c-

TltOIIOTIVE FOllCE.. . ••• .•. ••• . •• .. .. . ••..•..•••.•••••.... . ....•.•.•.. 199 IX.

ELECfaOLYTIC AND METALLIC CONDUcr1O:,;

X. XI.

THERMAL Al'lb VOLTAIC E~kO»OTI VE Fmt CF-'i ••• • •••••••• 246

XIJ.

MAGSETii.

•••••• • n8

MAGNETOSTATIC FIE LDS.. . ...... .. ........ , . ••• ••••• , 265

THE MAGNETIC FIELD OP TKE CO!o!OUCTIOS CURRENT • .• ••• 286

XIJI.

EL£CTROMAG!o!ETIC INDUCTION..

XI\' ,

UXJTS A.SO DUI ES!',,JOXS ... . .. .. ... . . .... . ... ........... . . .. . ... . .. . ... 415

XV.

•...........

CO!'l\'ECTION. A!-1D DtSf'LACEkEl'IT C URREXTS.

.. . ......... 414

ELECTRIC CUR REX T

XVI.

THE

Fwx

OP

ELECTkO)I AGXITTIC

WAVES. ... ... ..... .. . ......... ,....

332

TIU! Gtc:-.t:llAI.

ESE kG\'.

I::!.F.(.7 .KIC • ••

433

ELEMENTS OF ELECTROMAGNETIC THEORY. CHAPTER I. GENERAL ELECTROSTATIC THEORY.

1. Blectril.eatioa bJ Couact. Politin u4 •eraun Charrea. Let one end of an ebonite rod and a dry woolen cloth be rubbed

or strongly pressed together and then separated i and let a second rod and cloth be treated in the same way: The rubbed part of each cloth will be found, on trial, to be attracted toward the rubbed part of each rod, while the rubbed part of each cloth will be rq:,ellcd from the rubbed part of the other cloth, and the rubbed part of each rod from the rubbed part of the other rod. These arc examples of rleclric phenomena. The region in which they arc manifested is called an rlutn·c ji~ld (§ I 1), and the medium which permeates this region -air and a:ther in the above case- and through which electric influences are trans~ mitted is called a dulutn·c. The parts of the ebonite and wool nibbed together are said to be electrified, or to possess rlutN d/Jllrges. The two pieces of woolen cloth are said to have like dw-ges, since they were similarly treated and since what is ttpclled from one is repelled from the other, and what is attracted toward one is attracted toward the other. Similarly, the two cbonitc rods are said to have like charges. But the wool and the cbonitc are said to have unlilu or opposite charges, since what is rq,rlled from one is attracted toward the other.

ELEMENTS OF ELECTRO:\fAGNETIC THEORY.

Like ebonite and wool, any two different substances, or portions of the same substance in different physical conditions, exhibit electric properties after intimate contact and separation. One of the bodies behaves like ebonite rubbed with wool, the other like the wool. An electric charge like that of wool after contact with ebonite is called a positivt charge, and a charge like that of the ebonite, a ntgativt charge. The terms positive and negative are justified by the opposite properties of the two kinds of electrification, but there is no reason except convention and resulting convenience why the two terms should not be interchanged. In addition to the forces between electrified bodies, forces arc found to exist, in general, between an electrified body and an insulator(§ 2) not electrified (Chapters IV. and VI.). 2. Cond.ucion ud Inaulatora. Eleotri.8cation. bJ Cond.uotion.. A rod of ebonite electrified at one end exhibits electric properties only at that end; while a rod of metal, held by an ebonite handle and electrified at one end, becomes electrified at once {apparently) all over its surface. Substances like the metals, by which an electric charge is distributed with extreme rapidity, so as to come into a state of equilibrium within (usually) a small fraction of a second, are called tlcdni: condurtors. A body charged by connection with an electrified body through a conductor, like the far end of the metal rod mentioned above, is said to be cltctrijied /Jy ronduelwn. Substances like ebonite, over or through which an electric charge is transferred only with extreme slowness, are called electric insulators or nou-condudors. Among ordinary molecular substances pnfcct insulators and pt:rfcct conductors do not exist, no such substance completely and for an indefinite time preventing all transfer of electrification, and all offering more or less obstruction to such transfer. There is every reason to believe, however, that free rether (a" vacuum ") and clean dry gases containing no (electrolytically) dissociated molecules have the properties of a perfect insulator (Chapter IX .).

GE!'\ERAL ELECTROSTATIC THJ::OKY.

Among substances possessing high conductivity arc the metals, graphite, and salt or acid solutions ; among those with high insula.ting properties are (undissociated) gases, fused quartz (cold and in the solid state), ebonite, cold glass, silk, and wool. A substance which is an excellent insulator in one condition, hown-cr, may in another condition have the properties of a conductor. Thus cold glass is an excellent insulator, but as the temperature is raised its insulating properties disappear. Also, under very great electric stress the insulating propertic~ of all molecular substances break down. A body completely surrounded with insulators is said to be ias11/at,d. A conductor can be completely disdrarr~d by bringing it into contact at any o,u point "ith the inner surface of a hollow closed conductor(§ 4), such as the walls of the room within which the experiments arc performed, provided there arc no {insulated) clcctri6cd bodies within. When ~onnectcd to the walls of the room, or the earth, the conductor is said to be carllted. From an insulator the electrification can be entirely removed only by applying a conductor at every electrified point, ,. g., by immersing it in a conducting gas or liquid.

I. Blect:riAcation. by lnduotion. An insulated conductor, when brought near an electrified body, i. e., into an electric field, itself becomes electrified. Examined by the methods of § 1, the charges of the more remote and nearer ends of the conductor arc foand to be similar and opposite, respectively, to that of the original electrified body. A conductor electrified in this manner is said to be electrified by induction. If the conductor, while still insulated, is removed from the cltctric field, all signs of electrification disappear. But if, while llill in the ticld, it is connected with the walls, or eart/,cc/, the dtctrification similar to that of the original charged body disappears, while the opposite electrification of the near end remains. If the conductor is now insulated and removed from the original

4

ELEMENTS OF ELECTROMAGNETIC THEORY.

electric field, this charge becomes more evenly distributed ove its sutface (§ 42). ln this manner any nu111ber of conductor may be given charges opposite to that of a given electrified bod: without, as may be proved by the method of § S, diminishing o increasing the latter's electrification. '-8. Bzperlm.eW with Hollow CloNd Cond:a.cton. Eleotri Bcreeu. Let A denote an insulated hollow conductor having , closely fitting conducting lid, B, with an insulating handle. Le A be connected with an electroscope or electrometer (Chapte 111.), C, by means of which any change in the state of electri fication of its exterior (or interior) surface may be detected; an1 let A be kept closed except when another body is being intro duced into its cavity, or removed therefrom, or its position in side (or outside) altered. 4. (1) Let the elcctrometcr be placed outside of A. If Ai initially unelectrificd, and an insulated unelectrificd conductor, L is now introduced -into A without touching it, the inner and outc 1utfaces of A will ~omc electrified by induction (§ 3) will charges opposite and similar, respectively, to that of D. An, the electrification of the external surface, as indicated by th elcctrometer, will be found to remain absolutely unaltered how soever D is moved about within, even when it is brought intt contact with A; but D, on being insulated after contact, and tho removed from A's interior, will be found completely dischargec:i This process may be repeated indefinitely, D always becomini completely discharged on coming into contact with the inne surface of A. I( A is initially electrified in any manner, the phe nomena will be precisely the same, except that the externa electrification and the corresponding indication of the electrom cter will be different. (2) Let the elcctrometer be placed within A, either connecte, with A metallically, or insulated therefrom. In this case it wil be found that if there arc insulated charged bodies within A, th electrorrieter will give a. certain deflection ; that if there are n,

GENERAL ELECTROSTATIC THEORY.

insulated electrified bodies within A, the electrometer will give no deflection ; and that its indication in either case will remain absolutely unaltered howsoever the electrification of the exterior of A or of external bodies is changed, even if A is connected to the waUs of the room. These experiments arc due to Faraday, who constructed for the purpose of performing (2) a closed conductor large enough to enable him to make the observations while himself inside the cavity. An experiment similar in principle to those of Faraday, but bs general, performed earlier by Cavendish and repeated later by Max.well with all the precision of modem investigation, gave identical results. From the experiments just described it follows that, when there is electrical equilibrium, 1. A11 elatn'c cluzrge ca111t0I exist in llu n1bsta11u of a conWtw, or 011 IM i11ne, surface of a /,o//f.lW cloud co,,duc/or (unless there arc insulated electrified bodies within). For D, on being removed from A, of whose substance it formed a part, electrically, while in contact, was always unclectrified. 2. An electn'c field (§ I 1 ) · dots not trist wit/u"n tlu ltolltrdJ ef a dostd co,,dudor (unless there are charges inside). For in (z) the electrometer was unaffected (by induction or otherwise) no matter what the external electrification, except when there were insulated charges within. 3. TM elutn'c cltargrs a11d dec/n'c fields wit/tin and without a lwll'1W dosed collfiudor are absolutrly indrptndml of one nnolhtr. The conducting shell thus completely screens each of these regions from all static effects in the other. 4- An ~lcctric firld dots Ml r.risl withill tlu s11/Jstancr of a condlu:tor. See § I 5.

5. Bqul Charges. Two electric charges of the same sign are, by definition, of the same magnitude if they produce the same eff'ect on the electrification of the vessel A when introduced in succession separately.

6

ELEMENTS OF ELECTROMAGNETIC THEORY.

Similarly, two charges of opposite signs are, by definition, equal in magnitude if they produce no effect on the electrification of A when introduced simultaneously. These definitions are independent of the particular closed conductor A used, as two charges defined as equal by means of one such vessel are found to remain equal when tested in the same way with any other hollow closed conductor.

6. Politive and ••ratin Charge• are Alway, Developed li.m.ultaneouly in Equal Amount.I. lf two bodies electrified by contact are introduced into the vessel A simultaneously, the in. dication of the electrometer remains unaltered. If an electrified body is insulated within A, and if an insulated uncharged conductor is then introduced in addition, the latter becomes electrified by induction, in conformity with § 3, but the indication of the electrometer remains unaltered. In these cases, therefore, positive and negative charges are developed in equal amounts(§ s); and in the same way it may be shown that this is always the case, howsoever the elcctrifica. tion is produced. 7. The Total Quutity of Bleotrilcation. ii Unaltered by ConIf the two insulated bodies of the last experiment a.re brought into contact with one another while inside the vessel A, or if they are brought into contact with the inner surface of A itself, conduction occurs, but no effect on the external electrification is produced. From this it follows that when conduction occurs, the total (algebraic) amount of electrification is unaltered. Corollary. T/r.c cliargcs induced on //,r imerr and OU/tr s11rfacts of A wltcn an electrified body is introduced and ,insulatrd within, -as· in § 4, arr ca.di ef tlu same 11zar,,itude as that of 1hr i11su/atcd 5ody. For when D touches A, the charges of D and the inner surface of A completely disappear by conduction, since D on removal is unelectrified; thus their algebraic sum is zero. And the (opposite) charges on the inner and outer surfaces, being induced, must, by § 6, be equal in magnitude. duction.

CEXERAL ELECTROSTATIC THEORY.

7

L Blecbic Cba.rgea of Both Ii.D.da Ieuured. in Tenu of a IUl(le Arbitrary 1Jnit. In addition to the hollow conductor A of§§ 4-7, let there be provided another similar insulated vessel B, sufficiently large to admit A through its opening; and let the conductor D be given a certain charge (suppose positive for the sake of definiteness), which will be adopted as a provisional unit. Ir now D is brought within A and kept insulated, the outer surface of A will have unit positive charge. If A is brought inside Band then into contact with it, this charge will disappear, as will also the charge induced on R's inner surface, leaving the outside of B with unit positive charge. If A is now removed from Rs interior and then D from A, the negative charge induced on A's inner surface will pass to the outer surface and will disappear when A is discharged. This complete process may be repeated any number of times. Each time 8 will acquire an additional unit positive charge, and thus may be given a masured positive charge which is any integral multiple of the original unit. To give Ba negative charge measured in terms of the same unit, the outer surface of A must be brought into contact with the inner surface of a hollow closed conductor after the introduction of D , when the positive charge will disappear from the outside, leaving unit negative charge upon the inner surface. \\'hen D is removed, this charge will pass to the outer surface of A, and will be given up wholly to B when A is brought into contact •-ith B's interior. B will now have unit negative charge, and by removing A and repeating the process may be given any number of units negative charge desired. To obtain any submultiple, 1/n, of the original charge, it is only necessary to arrange symmetrically in contact the original conductor D and n - I precisely similar and equal Conductors, all other bodies, except the surrounding dielectric, supposed homogeneous and isotropic, being so remote as to have no appre. ciable effect. Then, by the principle of symmetry, each con• ductor will ta.kc 1/ 11 of the original charge.

ELEMENTS OF ELECTROMAGNETIC TH~RY.

8. The Law of Coulomb. Let two small spherical insulated conductors which can be given any charge desired, measured in terms of some provisional unit by the methods of § §S and 8, be so connected with a dynamometer, such as a gravity balance, that the force F between them can be measured as their charges, 'Ii and 91, the distance L between their centers, and the surrounding dielectric arc varied. Then it is found by experiment that, (1) However the distance L and the · charges 9 1 and q1 are varied, provided all the experiments are performed in the same dielectric, and provided that this dielectric is homogeneous and isotropic and extends to a great distance on all sides of the elccbified bodies, F is in the straight line joining the centers of the conductors; is directly proportional to the product of their charges, being repulsive (considered positive) when the charges are like and attractive (considered negative) when the charges are unlike, as already known from § 1 ; and the greater L in comparison with the linear dimensions of the charged bodits, the more nearly inversely proportional to L 2• (2) In different dielectrics, with all other.conditions the same, the force is different, and always less than in vacuo (free zther} The general expression for F. when the linear dimensions of the (not necessarily spherical) charged bodies are negligible in comparison with their distance apart, is therefore

(1') where c is a constant depending on the medium in which the experiments are performed, called its permillivily or didrdric cons/an/, and A is a positive constant depending on the units in which q1, 91 , L, F, and c are expressed. (1') expresses C,m/o,nl,'s law. The lLatioul Eleotroltatic Unit Charge. Unit PermtttiTitJ. In what follows, unless the contrary is stated, the centimeter will be used as unit length, the dyne as unit force , the permitti,•ity of free zther, which will be denoted by co> as unit permittivity,

GENERAL ELECTROSTATIC THEORY.

9

and as unit charge the charge which each of two indefinitely small bodies must have in order that when at a distance o( I cm. apart in a vacuum the force between them may be 1/4.,, dyne. This lfnit charge is called by its originator, Oliver Heaviside, the ruun,a/ rlrclrosJahC •nit c/,arg,, and c, is called the dutrostahC nil jln'Milmi/7. Methods of measuring permittivity are discussed in Chapter

VII. The conventions just made give, by the above equation, A - 1/4.,,., and the equation reduces to (1) which, in addition to being a particular case of(•'~ is a particular case of (2). The direct experimental investigation of the law of force is due to Coulomb, but is not capable of great precision. The law, as stated by Coulomb, is_ most satisfactorily established by the consideration that all experimental knowledge is in perfect accord with an electrical theory based largely upon the assumption that the laws expressed in (1) arc exact.• A reason for the law of inverse squares and a justification of the tenn rational unit will be given in §§ ), II., and 24. The dimensions of electric charge and the other electric quantities, as well as other systems of units, will be considered in Chapter XIV. For ralio,w/ r/rclro.rJatir the abbreviation RES will hereafter be employed. 10. If any one of the experiments described abo\'e is repeated in different dielectrics, the results in all cases will be identical, except that, in conformity with § 9, the force between two charged bodies will always depend on the surrounding dielectric. • The

c...t'Ddi5h

C'OIIUnOD

deduction ol the la" of invene ladeuer. If two cquipotentials of radii l,, = L 1 + d are replaced h>· infinitely thin conductors, the electric field will remain unaltered except that it will become discontinuous at the surfaces of the conductors. The charge upon unit length of the outer surface of the inner cylinder is now 'J, and that upon unit length of the inner surface of the outer cylinder is - q, and the two conducting surfaces with the intervening dielectric form a condenser whose field is given by (17) and ( 18). The charge upon the straight line and that on the inner surface of the outer cylinder together with all the external charges a~ electric images of one another in the inner cylinder, etc. The conducting substance may ht' extended into the regions within the inner surface and without the outer surface in any manner, or the fields of th_ese ,egions may be wholly destroyed, without affecting the field of the condenser. For the voltage between the plates of the condenser, ( 18) gives ~ and

V,- V,-q/2.-,J,.''dL/L-q/2,r, -log(1 +d/L,)

(20)

The capacity of a lenbrth A of the condenser is

S-qA/(V,- I';)- 2.-ul/log(1 +d/L,)

(21)

and the capacity per unit area of the inner cylinder is

S'

= S/2"£,A = c/L,log (1 + d/L,) - ,fd(I - }d/1., + ld'/L,' - .. ·)

(22)

ELECTRIC FIELDS ASD CONDENSERS.

67

1£ d lS kept con5tant and L 1 made to increase, the field normal toa gi\·en portion of the inner (or outer) cylinder evidently approaches uniformity; and in the limit, when L,, - infinity, any finite portion of the condenser becomes a parallel plate condenser (I 12) of capacity efd per unit area. In any case when d/l is small the capacity per unit area is approximately c/d. The energy of a length A of the condenser is

W- }qA(I~ - V,)- q'A/4.-e -log(1 + d/L,) _,,A,flog(1 +d/L,) (V,-V,)'

(23)

The field of an infinite isolated circular cylinder uniformly charged is given by the above equations on making L 1 infinite and ~ zero. v; - V, can be easily obtained by the direct application of the law of inverse squares. Let the field outside the condenser be zero (though the results obtained will be independent of this .wumption); then V,- o, and v; is the potential at any point on or within the inner cylinder, and is therefore the potential at any point Pon this axis. Hence, from the figure (Fig. 15),

----------- -k:u·- - - - - - - '

,,,,' '

P\t.15.

I~ - V,- V, -

2[

[q2.-L,dr/4.,,,(.r + L,')'

(a)

-q=L,tfr/4.,,,(,'+L,"f] = q/2.-e-log L,/L, as in (20) above. 10. La.plaoe'1 Bquti011 for the Cylindrically Radial Field. be obtained directly from Laplace's equation.

v;- V. can also

68

ELEMENTS OF ELECTROMAGNETIC THEORY.

Without transforming the general equation, we can obtain directly, by a simple process similar to that employed in I 3, tbe special form it assumes in a cylindrically radial field Thus we find

l.d'V/dL' + dV/dL - o

(14)

Hence, by integration,

LdV/dL-C} dV-CdL/L

(•)

V,-V, -C i"dL/L-gf:z,,,, logL,/L,

(6)

and

since when L-

£., (a) gives C- -LiJ ho Wnite Parallel llnighl LIDN or Ciroolar ConduotiJas CJlinden, with Clwgeo g ud. - q Oil Vnit Lnfth. Consider first two electrified straight lines, distant 2a apart, and cut by a perpendicular plane in the points A 1 and A, Fig. 26. By symmetry, the distnbution of the lines of displacement is the same in every such plane. More0\'tr, all the lines emanating fro'!l a point A 1 pus to the point A2 in the plane containing the two points and perpendicular to the two lines.

Flf. 26.

The potential at any point P distant L 1 from A 1 and L, from Alis

V- V, +

V,-g/2.-,(fdL /L - I:dL / L)

- g/2.-ci"'dL/L-gJ,.-c,Jog L,/L, This is a1so the equation of the section by the plane of the paper of the equipotential surface whose potential is I : Dy giving to V different values the corresponding surfaces may be obtained. The displacement at Pis

D- Vector sum of D1(-q/2"1rL 1 directed from A,) and D,( = 9/2'1rL 2 directed toward A,) and the intensity E is D/c.

(49)

84

ELEMENTS OF EUCTROMAGNETIC THEORY.

From (49) the equation or any line of intensity can be obtained by the method of § 1 S· Proceeding exactly as in that article, we find the equation

Bl+ 81- C- constant

(50)

which is evidently the equation of the arc of a circle terminating at A I and A,, and cutting perpendicularly the line normal to A 1A, at its middle point If i:he line whose equation is sought cuts the normal to A 1A 1 at its middle point O at a distance x from 0, we have, for this point, 81 = 81, and

C- 26, - 26,- 26- 2 co,-'[aJ(a' + ..-'JI]

(51)

The equipotential surfaces given by (48) are circular cylinders, or their lines of intersection with the plane A,A,P circles, orthogonal to the lines of intensity. For (48) may be written

(SZ) a constant for the curve, or surface, whose potential is V; and this is the equation of a circle cutting the line A 1A1 and ·with its center Con the line A 1A 1 produced. The radius of the circle whose potential is Vis

R-2l1a/(li'- 1)

(53)

the distance of its center C from A 1 is

A,C-Rh

(54)

and the distance of C from A1 is

A,C-R/h

(55)

To obtain the resultant displacement IY at Pwe must obtain the vector sum of D 1 and D 1 , Fig. 26, which \\-;U be along R normal to the equipotential. Since D 1 and D 1 are directed along L 1 and L 1 respectively, and since, by (49), D 1 / D 1 - I.,/ L 1 , the triangle whose sides are D 1, D 1 , and ./Y is similar to the triangle A 1PA,; so that

D' JD, - 2aJL,, and D' JD,- 2aJL,

ELECTRIC FIELDS AND CONDENSERS.

85

U- 2aD,/L,- 2aD,/L, -gaf'rL,L,

(56)

and lhe resultant intensity Eis equal to U /c. The force F upon a length A of either electrified line, considered positive when tending to increase a, is

F- -gA-g/e2.,2a- -,tA/4.,,a

(S7)

The plane diagram of the field, drawn by Maxwell's method, f 14, is given in Fig. 27 (from Webster's 11'ory of EJ"trinly a11d Jlap,tism, f 159). The tubes of displacement and the equipoltntws are mapped out by moving the diagram perpendicularly to its plane.

Pit, 27.

If for any equipotential surface the coincident surface of a conductor is substituted, the electric field on the side facing this surface v.,11 remain unaltered. The above field therefore includes, as particular c:a5C5, the fields bounded by (1) An infinite straight line and a parallel infinite conducting circular cylinder,

86

ELEMENTS OF ELECTROMAGNETIC THEORY.

(2) An infinite straight line and a parallel infinite conducting plane, (3) An infinite conducting circular cylinder and a parallel in, finite conducting plane, (4) Two parallel infinite conducting circular cylinders, internal or external (either or neither surrounWng the other), all with charges 'I and - 'I upon unit length. The fields of §§ 8--9 .ire particular cases of (4) when one of the two lines is removed to infinity. As systems of practical importance, we shall discuss (4) for the case in which the two cylinders arc external to one another, each of the same given radius R, with their axes at a given distance 2d apart, and charged to potentials V a'nd -V, and (3), which is a particular case of (4). To obtain the electric field terminated by the two cylinders, we must find the distance a and the charge 9 upon unit length of the positive cylinder. From the similar triangles A 1 CP and A 1 CP (Fig. 26) we have whence

(d + a)(d- a)= R'

(58)

a =(d'-R')I

(59)

From (54) and (55)

li-(d+a)/R-R/(d-a)- [d + (d'-R")']/R (6o) -R/[d-(d'-R')I] For the cylinder whose potential is - V we have

log k- 2rcV/9 and therefore

9- ,,,,V/logk- ,,,,V/log[ (d+ (d'-R")1}/R] (6 ,) - =cV/log[R/(d-(d'-R")'}] • From (4g) and (56) the field can be determined, by maldng u,e of(61), at all points. The capacity of a length A of the system is

ELECTRIC FIELDS AND COSDE.~SERS.

87

S•9A/2V- TcA/logk - TcA/log[{d+(d'- R')')/K] (62)

and the energy in the same length is IV-!9A•2V- ,.-,AV'/log[{d+ (d'-11")')/K] - j/A /.-c-log[{d + (d'- R')I) /K]

163)

If the infinite plane surface of a conductor is placed coincident 'Alth the surface of zero potential (the plane pas.-.ing symmetrically between the conductors) the field on the side facing the conductor "'ill remain unaltered ; it is simply half the field just considered. The capacity of a length A of the condenser formed by the Ulfinitc plane and the cylinder with the dielectric is

S, - 9Af V-2S

(64)

and the energy is half that contained in the complete field surrounding the two cylinden, or

(65) The force F acting upon a length of A of either conductor, plane or cylindrical, is given by (57). It can also be obtained by differentiating W1 with respect to d, or W with respect to 2d, by the method of§ 55, I. Thus

F- -9'A/4.-,a =_.,,A V'/(d'-R')llog[{d+ (d' -R')IJ /R]' IO. The Pield of & Line Doublet. parison with L1 and L.,, we have

When

2a

(66 )

is small in com-

V- 9/2wc·log L,JL,

-(approximately)9/2"'c •log {I+ 2acos8/K) - vf2,rc· 2acos 8/R(1 - 2acos 8/2R + .. •)

(67)

if R is written for L 1 and if 8 denotes the angle between L 1 and the lincA,AI' If now the product q 2a is kept constant while a is diminished indefinitely, (67) approaches the limit

88

ELEMENTS OF ELECTROMAGNETIC THEORY.

V- 2agcos8/2rcR-Mcos8/2rcR

(68)

where Mis written for 2ag. This system is called a li11~ do,d;,/,t, and Mis called the momnl of the doublet.

Ac. 211.

The radial and tangential displacements at a distance R from the doublet, at a point where R makes an angle 8 with the line .A,A 1, now infinitely short, arc

and

D, - - cdV/dR- A'1cos 8/2rR'

(69)

D,- - cd V/dT- - c/R-JV/J9- !,fsin 9/ttR'

(70)

The total displacement is equal to

D- (D,'+ D,')•- Aff2rR'

(71)

and makes an angle 28 with the line AiA 1 (the axis of the doublet). The lines of intensity are evidently circles tangent to the axis at 0, and the equipotentials circles perpendicular to the axis at

ELECTRIC FIE:LDS AND co:i-oE.~SERS.

89

0.

The plane diagram of the field is given in Fig. 28 (from \\·ebster·s 17,cory of Elutrinty aNI Mag,utisllf., § 44), the tubes of displacement and equipotential surfaces being supposed generated by moving the diagram perpendicularly to its plane. The method of drawing the diagram is easily understood from Fig. 29. Smee there is an infinite number of lines of displace-

f'1c.29.

ment within a circle of any finite diameter a, only the lines lying outside some such arbitrarily chosen circle can be drawn. The same is true of the equipotential tines. The flux through the tube between the cylinders of unit depth v.ith diameters a and J' is

n-[M/2,r-dy/y' - M/2,.. •(1/a- 1/;•) Hence by giving Il any set of successive values differing by a constant the diameters(;•) of the corresponding lines of displacement may be obtained. The voltage from the circle of equal potential of diameter b to the circle of diameter x is

V.-

i:-:_-Mf,,.., . (1/b- 1/.,)

Hence by starting with a circle of diameter /J and giving J~ - V any set of successive values differing by a constant, the diameters (.r) of the corresponding equipotential circles may be found. 1

go

ELEMENTS OF ELECTROMAGNETIC THEORY.

81. rho Bleotric Fiold Bumnmolins an I1Glatecl Co11ha11Jir Spheroid. First it will be shown that within the region cnc109ed by a homogeneous material shell whose surfaces arc similar and similarly situated ellipsoids there is no gravitational field of force. Such a shell is called an tllipsflida/ lum,tzuid, Let a cone, Fig. 30, of infinitesimal angle d• at any point A" in the region cut from the shell the volumes H'C" and IY'E". If p denotes the density of the shell, g the gravitation constant, L the distance from A 11 of any clement of volume th of the shell, the intensity at A" in the direction A"C" due to the masses in B"C" and IY'E" is

dG-f

£.::~pdT/L

1

-g

1:::•~/L

1

since the attractions due to the masses in B" C'' and D''E'' a.re in opposite directions. Since th - L 2dl!HiL, the integrals reduce to

dG-rpd tector,

8. Blectro1aopu and Bleotrometen. An rlutrm,u/n- is an instrument for measuring voltages, or electric potential differences, by means of the forcive acting between electrified bodies. Other instruments for measuring voltages will be descn'bcd later. An electroscope is a crude electrometer, used principally for detecting rather than measuring electrical effects. ,. fte X:elrin Absolute Electrom.eter. This ·instrument consists of a condenser constructed Uke the standard parallel plate condenser of § 2 with certain modifications and additions : The plate B (in one of the commonest fonns of the instrument) is conne and V,. therefore positive, is shown approximately in the figure (all the lines of intensity should touch the conductors nonnally). Except near and beyond E and H, and G and F, the field is cylindrically radial, and its capacity per unit length is constant and equal to

v.

v.,.-

v.

S-2.-c·logL.fL, By symmetry, there is no resultant horizontal force acting on C. In general the vertical forces acting on C at H and E arc

CONDENSERS AND ELECTROMETERS.

129

not equal and opposite. The resultant force can be found as follows : Imagine C to be moved downward an infinitesimal distance dr. The capacity of the condenser AC is increased by Sd..r and that of the condenser BC by - Sdx, no sensible change occurring in the capacity of the non-cylindrical parts of the field. The increase in the energy of the field, if the voltages are kept constant, is

dW-Sdr(VJ- V,")

s

Hence, by § S, I., the resultant force acting downward upon C is

F-dW/dr-lS(VJ- V,')~iSV.,(V,+ V,) - SV.,( V,

+ ½V.,)

v.

(6)

v......

(1) If is great in comparison with the voltage to be measured, this C9Uation becomes, with a negligible error,

F-

sv..... - Va v... varied, F

Jicncc, if Va is kept constant and to

v_.. . .

(7) is proportional

(2) If A and B are connected to the terminal plates of an auxiliary battery consisting of an even number of similar voltaic cells in series, and if one tenninal of the cell, condenser, or other agent whose voltage V is to be measured is connected to the central point of this auxiliary battery, the other to the conductor C, we have, if 'I' denotes the e.m.f. of the auxiliary battery, V.,-'1', V,- V-i'V, V,- V+i'l'; and(6)bccomes

F-S'1'. V

(8)

so that Fis proportional to V if 'I' is kept constant. (3) If B and C are connected together, V_. - o, VA - V,u, and (6) becomes

(9) EquatiOJls (7), (8), and (9) indicate three methods of comparing voltagC5 with the instrument, the force F being meas-

I 30

ELEMENTS OF ELECTROMAGNETIC THEORY.

urcd with the baJance (or other form of dynamomcter). If Sis determined from direct measurement, and F measured in dynes, the third method gives an a!Jso/Ntt determination of the voltage VAA• The first and second methods are called lu-tut1slatic, as an auxiliary voltage, V,. or 'I', is employed in addition to that to be measured. The third method is called idiostatic, since the volt• age to be determined is the only one applied. 15. The )[elrin. Qu&dnn.t Eleetrometer. This instrument (Fig. 52) is constructed as follows: A right circular cylindrical me-

w

F1c-52.

tallic box, with its axis vertical, is cut symmetrically into four quadrants A, A', B, B', separately insulated on glass rode;, but connected by wires in pairs, A to A' and B to B 1 , so that when the field is static there is never a potential difference between op• posite quadrants. A light aluminium needle C, consisting of two equal opposite flat quadrantal arcs CC and C 1 C' attached by thin radii at their extremities to a central vertical rod R, is sus• pended from a support by two silk fibers (or other insulating torsion device) in such a way that the arcs CC and C'C' are horizontal, concentric with the quadrant cylinder, and midway between the top and bottom of the box. \Vhen the quadrants and the needle are all connected together, so that there is no potential difference between any two parts of the system, the arcs

CONDENSERS AND ELECTROMETERS.

131

C, C' are adjusted to lie symmetrically with respect to the two quadiant pain AA' and BB', as shown in the figure. To the rod R is attached a mirror M, by means of which and a la.mp and scale or telescope and scale any deflection, 8, of the needle an be read, and on the other side of the quadrants a vertical platinum wire JV, ending in a platinum vane J~ The end of the Tire and the vane hang free in dry sulphuric acid contained in a glass vcs5CI G, the outer surface of which is partly covered with ti\ foil The sulphuric acid serves to make electrical contact 'lrith the needle, to dampen the needle's motion,and to form with the tin foil and glass vessel a condenser of considerable capacity, whose: function is to keep constant the potential difference be· t1t.·ttn the needle and the case. The whole instrument is enclosed in a tight case, often an extension o( the vessel G (whose tin foil covering is then outside) and is kept dry by the sulphuric acid within. The case, largely metal, serves also to screen the needle and quadrants from any external field. If the instrument is symmctricaUy made and adjusted, the arcs CC and C' C' form with the two quadrant pairs AA' and BB' two condensers, the capacity of each of which, per unit angle subtended at the center of the system, is the same, let us say S, and constant, except near the edges of the arcs and quadrants, for all but exceedingly large deflections of the needle. Also, if the instrument is symmetrically made and in adjw~tmmt, the needle will obviously not be deflected, even when charged, as long as the quadrants arc all connected together. If the needle and the quadrant pairs AA' and Ell' arc charged, the nttdle will, in general, be deflected, coming to rest when the angle deflection, 9, is such that the torque T upon it due to the electrical stresses is balanced by the return torque due to the twist or the suspension. To find the relation between the deflcc• tion and the voltage, we may proceed a~ fo llows, using the method of§ SS, I.

or

132

ELEMENTS OF ELECTROMAGNETIC THEORY.

Let Va denote the voltage from the quadrants AA' to the quadrants BB', i-:. the voltage from the needle .to AA' , and v. the voltage from the needle to BR'. When 9 is increased by an amount d8, the capacity of the condenser formed by CC' with AA' is increased by Sd8, and that of the condenser formed by CC' with BB' is decreased by the same amount. The increase in the energy of the two condensers is then

dW - lSd8 V/ - ½Sd8 V,' - lSd8( V, - V.X V, + V,.) - ½Sd8 V,.( v. + V,) - Sd8 · V,.( v. - j V.,)- Td8

-Klld8 since v. - VA + v..._., and since Td 9 = K(J d 9 is the work done in twisting the bifilar (or other) suspension through the angle dfJ by the torque T of the electrical forces, K being the constant of torsion of the suspension. The last equation gives

T/K - 8 - 1S/K · V,.(V,

+ V,) = S/K · V,.(V,-½V,.) (to)

(1) If VA and Va -V.. + V"a arc very large in comparison with VA•• the voltage to be measured, V.ta may be neglected without appreciable error in the expression ( v. - ½V.Aa), and 8 is sensibly proportional to i-:..., and to V..,. Hence by making Vi, large, even small potential differences v... may be measured with accuracy. In this case ( 10) becomes

V,. -K/SV, · 8

(ll)

(2) If the needle is in metallic contact with one of the quadrant pajrs, as AA', v.A = o, v.., = VA•• and (10) becomes

V,}

= 2K/S - 6

(12)

Since the deflection in this case is proportional to the square of the voltage, alternating as well as steady voltages can be measured; but low voltages, either steady or alternating, cannot be measured with accuracy (except with very sensitive instruments). (3) The quadrant pairs AA' and BB' are connected to the terminal plates of an auxiliary voltaic battery consisting of an even number of similar cells in series, and one pole of the voltaic cell

CONDENSERS AND ELECTROMETERS.

I 33

or other· agent whose voltage V is to be measured is connected to the needle C, the other to the central point of the auxiliary battery. Then, if 'I' denotes the e.m.f. of the auxiliary battery, V,. - 'i', V, - V- i'I', V, - V + i'I'; and (10) becomes

V-K/~-8

.

(13)

In this arrangement the de8ection is a«11ral1/y proportional to V. whether Vis large or small in .comparison with 'I'. The first and third methods of using the instrument, in which a supplementary voltage is employed in addition to that to be measured, arc called lutcrostatic methods; the second is called idiostatic. \Vhile the quadrant clcctrometcr cannot be used for absolutt measurements, the factor multiplying 8 being impossible to determine with accuracy directly, this factor can be determined in any case by measuring the de8ection produced by a httrJ.Jn voltage, such as that of a standard cell. 7. Condemen ill Kultiple. When any number n of condensers whose separate capacities are S 1, S" • • ., arc connected in

s.

~

~

,. Flt. :IJ.

multiple, as in F'ig. S3, a, a compound condenser is formed whose capacity is

s-s,+s,+- --+s.

134

ELEMENTS OF ELECTROMAGNETIC THEORY.

provided that the field of each condenser is included, practically, between its plates only, and therefore does not affect appreciably the fields of the other condensers. For if Vis the common voltage between the .scpante pairs o{ plates of the compound condenser, 9 the total charge on each compound plate, and 9" 9,. • • • , 9. the charges on the sepante plates when the _condensers are charged separately to the voltage V, we have, for such a system, 9 - 91

+ 9, + .. . + 9.

and

S-g/V-(9, + 9,+ ··· +9,)/V -(S,V+ s,v+ · ·· + s.V),'V -s,+s,+ ···+s. 8. Condeuen in lerie1. When ,. condensers of individual capacities S 1, S1 , etc., arc connected up in series, as in Fig. 53, 6, a compound condenser is formed of capacity

S-1/(1/S,+1/S,+···+•iSJ

(15)

provided that the plates of each condenser are so close together that sensibly all the tubes from one plate terminate upon the other. For if Vis the total potential difference between the terminal plates of the compound condenser, q the (numerical) charge on each of them, and q 11 921 etc., and v;, V~. etc., the charges and voltages of the individual condensers, we have

since the intermediate plates arc all charged by induction, and sensibly all the tubes from one plate of each condenser terminate upon the other. Also V- v; + V,+ ···+ V:, Hence

S - 9/ V -9/( V, + V, + · · · + V:)- 9/(9 iS, + 9/S, + · · · +g/Sj from which ( 1 5) follows on cancelling q.

CONDENSERS AND ELECTROMETERS.

t loal Blecstrostatio

135

••tllod.a of Compariar CapaoitiN.

In ea.ch of the following methods the capacity of the elcctrometer, or elcctrometers, and connecting wires is supposed to be negligi>le in comparison with the capacities to be compared, or else to be included with them. (I) The capacities S1 and S1 to be compared arc connected in series with a battery of electromotive force V, and an elcctromctcr is connected across the plates of each ; or an elcctromctcr is connected across the plates of one, for example 5 1 , and another, with the battery, across the terminal plates. In the first case we have

s,v;-s.v.

whence

and in the second cue

S,/S,- V,/V,

(16)

S,V,- V/(1/S, + 1/S,) whence

S,/S,- V/V,-1

(17)

If the leakage and absorption (Chapter VI.) of the condensers arc negligible, the two measurements may be made in succession v.,ith a single clcctromctcr. (2) The capacities to be compared are arranged to be put in multiple by a switch K With K open let 5 1, to whose plates the quadrants of an electromctcr arc connected, be charged to a voltage V. and then connected in multiple with S 2 , when both condenser.i wilt come to voltage _V,. Then we have

S,V~ (S, + S,)V, whence

S,/S, -

VJ V,-

1

(18)

(3) In this method the condensers whose capacities 5 1 and 5 1 are to be compared are charged in multiple to the voltage 1': insulated, and then again connected in multiple, but in such a way that the positive and negative plates of I are connected to

136

ELEMENTS OF ELECTROMAGNETIC THEORY.

the negative and positive plates of Immediately after charging

2,

the final voltage being Yr

After the final connection in multiple

9, -9,-(S, -S,)V-(S,

+ S,)V,

Hence

S,/S,-(V- VJ/(V+ VJ It is obvious that the above three methods cannot be applied when one or both of the condensers arc of the guard ring type, thus ha\'ing more than two conductors. The following method of testing the equality of the capacities of two guard ring condensers was devised by Maxwell. It can also be applied when only one, or neither, of the condensers is of the guard ring type. In the last case it becomes identical with the last of the preceding methods, which is an extension of a method due to Cavendish. IO. Xu.well'&Xethod of Testing the Equality of the Capacity of a Guard Ring Conden..er and that of any Other Condeuer. • Let

A be the disk, B the guard ring and sheath, and C the larger plate of one of the condensers; and let A', B', and C' be the corresponding parts of the other. If either condenser, as ABC, is of the simpler form with only two conductors, we have only to suppress B and to suppose A and C to be the two conductors, it being understood that sensibly all the tubes of induction pass from one plate to the other when the condenser is charged, Let B be kept always connected with C', and B 1 with C. Then ( 1) Let A be connected with B, and C1 with/, the positive (for the sake of definiteness) terminal of a battery or other source of electrification, the other terminal of which is connected to • Mazwell, Trtatiu, § 229.

CONDENSERS AND ELECTROMETERS.

137

earth; and let A' be connected with B' and C and with the earth. The two condensers are now charged oppositely, so that A is positive an·d A' negative, and the 6eld of each is sensibly confined to the region between the plates. (:?) Let A, B, and C' be insulated from/. (3) Let A be insulated from B and C', and A' from B' and

C.

(4) Let B and C' be connected with B' and C and with the earth. The charges on A and A' remain unaltered in magnitu outward across a closed surface S surrounding any number of conceutrat,d magnetic poles of total strength m is equal to ,n, • This follows for an infinite or finite region cont.aining a homogeneous isotropic medium (p. - constant. except within the magnets, whose vol• With the quali6c:atiou made be]OII', In every cue, howe,·tr, the tl,u t1f 111,1.:,·· ahtir itc1n11U7 outward acrou I closed surface surrounding I rn l pole diMributcd in CJJ manner, multiplied by the i11ductivit7 of the inedium sulTOunding the magne1, ii eqDAl to the lltrtlllfth of the pole. Fw the an,.loeous electric case see VI. ud 1 V.

272

ELEMENTS OF ELECTROMAGNETIC THEORY.

umcs are supposed negligible) from (2), (3), (5), and (6), and a prottss of reasoning exactly analogous to that employed in establishing the corresponding theorem in electrostatics. The theorem holds only for concentrated poles, the flux from a distributed pole in general, very different from ,,,_ (See § 8, VI., for the corresponding electric case.) TIie .rtrrngtJ, of a tulH ef inductit111. It follows from Gauss's theorem exactly as in electrostatics that the ftux across ever. diaphragm of a given tube of magnetic induction in a homogcne, ous isotropic medium is the same. The magnitude of this ftux is called the Slrt1«gtl, of the tube. A 11,aJ trdJr is a tube whose strength is unity. \Ve here assume, as in strict accord with experiment. that the atrength of a tube is constant throughout its length whatever media it may traverse, whether the field is a pure magnetic fieJd or an electromagnetic field (XII.). For a pure magnetic field (in which there is no intrinsic magnetisation) this result follows from considerations exactly similar to those adduced to establish the analogous proposition in electrostatics, except that two concentrated permanent magnetic poles, one in each medium, must be cmplorcd instead of the closed electric condenser. In deriving Gauss's theorem the (infinitely small) volumes of the magnets and all their contents were neglected. It must alwars be remembered, however, that. as stated in § 2 , magnetic poles arc the analoi;:ues of the poles of elcctrets, not of true electric charges. \Ve mar. for convenience, consider only the flux fnm1 tor to) a pole in the surrounding medium, as we ha\'C just done; but we must remember that the same quantity of flux which emanates from a magnet at its positive pole enters the magnet again at its negative pole. making the total flux across any closed surfac~ s urrounding a single pole {or any number of poles) 7.Crtl. That this statement is correct for a magnet ";th concentrated poh..-s follows from Gauss's theorem and the fact that if any magnet is broken across its axis into any number of pieces, each

being,

MAGNETS.

MAGNETOSTATIC FIELDS.

z73

piece is a magnet with its positive and negative poles equal in strength (numerically) and pointing in the same directions as the corresponding poles of the original magnet. When both the original magnet and these pieces are very long and thin, as they must be to have approximately concentrated poles, the pole strengths of the pieces and the original magnet arc sensibly equal. We here assuu,r that all tubes of magnetic induction arc closed like the tubes just considered, whether in a pure magnetic field or in an electromagnetic field. This assumption is in strict accord with experiment and with a more general definition of magnetic induction given in Chapter XIII. Thus there is nothing in magnetism analogous to the discontinuity of electric displacement, or true electric charge. · Applying the above results to the element of volume at a p:,int, we get, obviously, div B -o-div µ.II

(7)

the flux into any element of volume across a part of its surface being equal to the flux out of the volume across the rest of the surface. lt. •arn•tomotin Force or Oa11uage. •&gnetic Potential ud Equipotential Surface ■ . The line integ ral of magnetic intensity, H cos 8 dL, along a path L from a point P 1 of a magnetic field to a point P 1 is called the magnrtomoh·vr forcr (m.m.f.) or rtUt.Ssagr from P1 to P1 along the path L. When this integral is the same along every path from P. to P, it is also called the differrncr of magnr/li: potrnlial between P. and Pt• or the fail of 111ag-,,rli'c potential from P 1 to P 1• From a process of reasoning similar to that of § 17, I., this is evidently the case in the field of a magnet (unaccompanied by electric currents). The unit gaussagt is the gaussage which exists between two points when unit work must be done to transfer a unit magnetic pole from one to the other.

J

274

ELEMENTS OF ELECTROMAGNETIC THEORY.

The fall of magnetic potential from a given point P to a point at an infinite distance from all magnetic poles is called the 1nag1uJic j>(llrnJial at P, and will be denoted by 0. This symbol •ill also be used to denote m.m.f. in the more general case. A surface which is everywhere nonnal to the magnetic intensity is called a magnetic